While "best" is subjective and depends on individual preferences, one recommended K-drama for 13-year-olds is Sassy Go Go / Cheer Up.
Sassy Go Go / Cheer Up is a high school drama, making it relatable for a 13-year-old audience. It focuses on themes of friendship, teamwork, and navigating the challenges of adolescence. The series avoids overly mature or explicit content, making it age-appropriate. Other dramas with good stories that could be considered are While You Were Sleeping.
Therefore, consider Sassy Go Go / Cheer Up (and exploring summaries of While You Were Sleeping to see if it suits your preferences) as a potentially engaging and appropriate K-drama choice.